# How to prove the formula of energy spectrum of strain rate in turbulence?

From Pope's turbulence book, for homogeneous isotropic turbulence, how to prove the following relation for strain rate energy spectrum

$$\langle S_{ij}S_{ij} \rangle = \int_{0}^{\infty} k^2 E(k)dk$$

where $$S_{ij} = \dfrac{1}{2}(u_{i,j}+u_{j,i})$$ and $$E(k) = 2\pi k^2 \Phi_{ii}(k,t)$$ where

$$\Phi_{ij}({k},t) = \dfrac{1}{(2\pi)^3} \int_{\mathbb{R}^3} e^{-i{k}\cdot {r}} R_{ij}({r} ,t ) d{r}$$ and

$$R_{ij}({r},{x},t) = \langle u_i({x},t) u_j({x+r},t) \rangle$$

• Two quick questions: (1) is the notation $Q_{i,j} = \partial_{j} Q_{i}$ and (2) in the $E(k)$ formula, are the indices of $\Phi$ supposed to be the same or should it be $ij$? – honeste_vivere Feb 2 '18 at 15:29
• As a hint, the Fourier transform of the autocorrelation is the power spectral density, by definition. – honeste_vivere Feb 2 '18 at 16:02